Chapter IV: Feedback sensitivity of high-coherence Si-III/V lasers
4.1 Optical feedback sensitivity review
We consider how undesired but unavoidable optical reflections affect single mode laser resonators where a single lasing mode exists with a large side mode suppression ratio. For lasers with two or more competing modes, or degenerate modes, as is the case for ring lasers, the main effect of optical feedback will be to break the degeneracy and cause one of the two competing modes to become dominant [76].
We therefore consider lasers designed around single mode resonators such as the mode gap resonators described in Chapter 2. We assume that the gain margin of the lasing mode is not significantly affected by the small level of optical feedback.
In the case of single mode resonators, the main effect of optical feedback can be modeled as changing the effective reflectivity of the output mirror [21] as shown graphically in Figure 4.1. For the case of weak feedback, the effective reflectivity of
Figure 4.1: The effective reflectivity of a laser in the presence of an external reflector. The source of the partial reflector can be either a small piece of dust, a bad connector, an optical component with a finite reflection, a bad fiber splice, or even an unconnected fiber.
the resonator’s output mirror (reff) is given by the sum of original reflectivity of the mirror (rL) and the reflectivity of the external reflector (rext) viewed at the position of the laser’s original mirror and can be written as
reff =rL +rext
1− |rL|2
. (4.1)
The reflectivity of the mode gap resonator evaluated at the frequency of the lasing mode can be found by considering the resonator as possessing two frequency de- pendent mirrors on the output side. The (complex) reflectivity of the two mirrors of our modegap resonator can be computed using the coupled mode equations by simulating the resonator as two halves each containing half the distributed defect as shown in Figure 4.2. The group delay can be found by solving the coupled mode equations [6] for each half the resonator. For our resonators, described in Table 2.1, the round trip time in the cavity is found to be approximately 6 ps, roughly equivalent to the round trip time of a Fabry-Perot cavity the length of the distributed defect.
The advantage of the mode gap resonator is that the mirrors are of extremely high reflectivity and the nature of the distributed defect dictates that only a singlehigh quality mode may exist. The reflectivity of the mirror as a function of the mirror length is found in the same simulation and summarized in Figure 4.3. Due to the phase matching condition of the mode gap resonator, the lasing mode exists at an offset when compared to the center of the band gap which necessitates the use of the coupled mode formalism to correctly estimate the reflectivity of the mirror at the lasing wavelength. If one assumes that the reflectivity of the mirrors is given by R(νL)= tanh2κL, the external quality factor of the resonator can be overestimated by as much as a factor of ten (10), as illustrated in Figure 4.3.
The lasing frequency can be determined by requiring a phase retardation of some multiple of 2π per round trip for a frequency within the band gap of the grating,
Figure 4.2: The simulation strategy to find the reflectivity as well as the round trip time inside the mode gap resonator, two necessary parameters that help understand the feedback properties of the resonator. In the top figure, the full donor mode resonator is shown with the conduction band in blue and the valence band in red.
The location of the optical mode with respect to the band edges is shown in black.
Figure 4.3: The reflection (R) and transmission (T) coefficient of the mirror portion of the resonator when the frequency is resonance with the mirrorδβ =0 and when the frequency is offset by 93 cm−1for a grating with κ =110 cm−1.
taking into consideration the complex frequency dependent reflectivity of the two gratings. The quality factor of the high-Q resonance can be found by using the group delay (τRT) of the mode and the transmission (T) or reflection (R) evaluated at the resonance frequency (ν0):
Q =2πν0·τRT· 1
−lnR(ν0) ≈ 2πν0τRT 1
1−R(ν0) =2πν0τRT 1
T(ν0), (4.2) For high-Q resonators, R approaches unity and it is therefore more convenient to describe the quality factor as a function of the transmission (or external loss).
Regimes of optical feedback
Now that we have established the tools necessary to analyze the passive resonator of the laser we are ready to analyze the properties of the laser in the presence of external optical reflections. Previous studies of optical feedback identify five (5) distinct regimes of optical feedback [77]. The first regime corresponds to small levels of optical feedback and gives rise to small changes in the lasing frequency while causing the linewidth of the laser to narrow or broaden depending on the phase of the optical reflector. In the second regime, two modes meet the phase matching condition and mode hopping is often observed. In the third regime, mode hopping is suppressed due to gain saturation and the laser oscillates in a single wavelength.
This regime is found to occupy a small range of feedback power ratio and as such is often ignored. The fourth, and most detrimental regime occurs at moderate levels of optical feedback where multiple modes can lase simultaneously regardless of the phase of the optical reflector causing the linewidth to suddenly broaden dramatically.
The fifth regime is that of an external cavity laser, where the system can be thought of as a large laser with a small gain section.
We will focus the discussion on the fourth regime known as the coherence collapse regime. In this regime, the location and the phase1of the reflector do not significantly affect onset of coherence collapse. For many lasers, the onset occurs quickly at relatively low feedback fractions near −40 dB [77]. One proposed strategy has been to increase the reflectivity of the quarter wave shift DFB grating by increasing the reflectivity of the mirrors in a quarter wave shift DFB laser (by increasing the productκL) [21]. Unfortunately, doing so often exacerbates any issues arising from spatial hole burning [78] and can decrease the absolute output power of the laser as it becomes increasingly under-coupled. As discussed in Section 2.3, if the quality
1The phase of the reflection is determined in part by the location of the reflector to within a fraction of the wavelength.
factor of the resonator becomes limited by the intrinsic quality factor (Qext >Qi), a rapid decrease in the output power is observed. It is therefore necessary to increase theintrinsic quality factorof the passive resonator before increasing the reflectivity of the external mirrors.
The feedback sensitivity of lasers is summarized by the feedback coefficient C = τext
τRTrext1− |rL|2 rL
p
1+α2, (4.3)
where τext is the round trip time of the reflector, α is the linewidth enhancement factor [17], [79],rextis the reflectivity of the external reflector, andrLis the complex reflectivity of the laser mirror at the output facet. AsrLapproaches unity, as is the case for high-Q resonators, Equation 4.3 becomes
C = τext
τL rextTLp
1+α2, (4.4)
where theTL = 1− |rL|2. In this form, it is clear that decreasing the transmission from the output facet has a crucial impact on the feedback coefficient of the laser.
The phase matching condition for the case of weak feedback is given by
∆φL(ν)=2πτL(ν−ν0)+rext1− |rL| rL
p
1+α2sin(2πντext+arctanα), (4.5) where∆φL is the phase matching condition of the laser in the presence of optical feedback, ν is the frequency, and ν0 is the lasing frequency in the absence of any optical feedback. A graphical representation of how the phase matching condition changes with increased feedback is shown in Figure 4.4. NearC =1, the two modes satisfy the phase matching condition and will start to compete. For even larger values of C, the number of modes starts to increase giving rise to competition between multiple modes. One important distinction between the small feedback regime, and the regime of large feedback that causes coherence collapse is that the onset of coherence collapse is relatively independent of the location of the reflector. This is because in the case of coherence collapse with largeτext, a multitude of modes with a separation of 1/τextexists. The mode separated from the original lasing mode ν0 by the relaxation resonance of laser will compete with the main lasing peak causing a dramatic increase in noise. As such, coherence collapse is typically characterized by measuring the relative intensity noise of the laser and observing a large increase near the relaxation resonance frequency.
Figure 4.4: Phase matching condition as a function of the lasing frequency offset without feedback(ν0)asC is increased. Plots are shown for α= 4, τext = 100 ns, Qext =1×105.